How can I create a pure black-white brick texture node?
You need to understand how UV coordinates work for this. Every vertex has such a pair of two numbers that get interpolated across the mesh. Suppose you have a square made out of four vertices with the shown UV coordinates:

The bottom left of your image will always get mapped to (0.0, 0.0) and the upper right to (1.0, 1.0). If the UV coordinates exceed these bounds, the image will get tiled, which means copies will get mapped to (1.0, 1.0) -&gt; (2.0, 2.0) etc. This means, you can simply multiply (scale) the the UV coordinates by using a mapping node and your texture will get projected onto the mesh multiple times:
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The only thing missing is the shift. For this, you want to copy your mapping node and input some value in the location x field. Then, create a MixRGB node which will choose when to use which mapping depending on some parameter. To create this parameter, you want to devide the scaled V coordinate by 2, take the fraction of it (which bastically is the rest when devided by 1) and see at which points it is greater than 0.5:

All that is left to do is to plug this output parameter into the MixRGB node, connect the two mapping nodes to it, if you haven't done so already and change the location x input of the second mapping node to your liking.